We propose a wavelet-based spectral method for estimating the (directional) Hurst parameter in isotropic and anisotropic non-stationary fractional Gaussian fields. The method can be applied to self-similar images and, in general, to d- dimensional data that scale. In the application part, we consider denoising of 2-D fractional Brownian fields and the classification of the clouds/temperature satellite images. In the first application, we use Bayesian inference calibrated by information from the wavelet-spectral domain to separate the signal, in this case the 2-D Brownian field, and the noise. For the classification of geophysical images we first estimate directional Hurst exponents and use them as an input to standard machine learning algorithms